258 problems

Let $(X,L)$ be a polarised smooth projective variety over $\a9c$ of complex dimension $n$. An extremal metric is a Calabi extremal Kähler metric on $(X,L)$, and $(X,L)$ is relative…

Let $X$ be a Fano variety. The invariant $b3(X)$ characterizes a notion called Gibbs stability, while $b4(X)$ is the stability threshold in the log Fano setting. Gibbs stability co…

Delta-invariant conjecture. For every $n$, one has

Let a polarized variety mean a projective variety equipped with a polarization, and consider the moduli space of K-polystable polarized varieties. K-moduli conjecture. A set of K-p…

Let $Y\to B$ be a holomorphic submersion equipped with the relevant relatively symplectic structure. An optimal symplectic connection is a connection satisfying the optimal symplec…

Tian's conjecture. K-stability implies the existence of a Kähler–Einstein metric on $M$.

Let $(X,L)$ be a polarized smooth projective variety, where $L$ is an ample $\mathbb{Q}$-divisor. The pair $(X,L)$ is K-polystable when it satisfies the algebro-geometric K-polysta…

Let $X$ be a Fano manifold with Picard rank one. A Fano manifold is K-semistable when its Donaldson–Futaki invariant is nonnegative for every test configuration. Odaka–Okada's conj…

Let a Fano fibration be a fibration whose fibres are Fano varieties, possibly with singular fibres in a degeneration. Fano-fibration degeneration conjecture. A Fano fibration is po…

A Fano variety is a variety with ample anticanonical divisor. A normal test configuration is a normal test configuration of the Fano variety, while a special test configuration is…

Let $y\in Y$ be an $n$-dimensional klt singularity that is not smooth. The Gap Conjecture. normalized volume satisfies … and equality holds if and only if $y\in Y$ is an ordinary d…

Let $F$ be a number field, $(X_{\eta},L_{\eta})$ a normal polarized projective variety over $F$, and let $(X,L,h_L)$ range over all metrized polarized normal models over…

Let $(X,L)$ be a polarized variety and let $(d4c^1)^{\mathrm{NA}}$ be the space of non-Archimedean finite-energy metrics, with non-Archimedean entropy functional…

Let $M$ be a toric manifold with moment polytope $P$, and consider the real moment map equation for a torus-invariant complex-structure deformation. The preceding calculation shows…

Let $X$ be a Fano manifold. The K-stability–Gibbs stability conjecture. $X$ is (uniformly) K-stable if and only if $X$ is (uniformly) Gibbs stable. Here uniform Gibbs stability mea…

Let $I$ be a chiral ring, and let $X=\operatorname{Spec}(I)$ be its associated variety. The variety $X$ is K-stable when the Futaki invariant $F$ is nonnegative for every test symm…

Let $X$ be a $\mathbb{Q}$-Fano variety with $\delta(X)\leq 1$, and let $0<\mu\leq 1$. The notions of $\mu$-twisted K-stability and $\mu$-twisted uniform K-stability are the propert…

Optimal Destabilization Conjecture. If $\delta(X)\leq 1$, then there exists a divisor $E$ over $X$ computing $\delta(X)$, that is,

Toric Yau–Tian–Donaldson conjecture. For every $f\in\mathcal{T}(P,\Lambda)$,

Fujita–Odaka conjecture. Uniform K-stability is equivalent to

Modified K-stability conjecture. The modified K-stability condition should be equivalent to the existence of an extremal metric in the polarisation class.

Let $(X,L)$ be a polarised variety, let $T$ be a maximal torus of automorphisms of $X$, and let $F_\chi(\alpha)$ denote the modified Futaki invariant associated with the extremal v…

Let $X$ be a polarised smooth projective variety. K-stability conjecture. K-stability is equivalent to the existence of a Kähler–Einstein metric, or more generally, a Kähler metric…

Let $(X,L)$ be a polarized manifold. Let $\mathcal{H}$ be the space of Kähler metrics in the class $c_1(L)$, let $\mathcal{H}^{\operatorname{NA}}$ be the space of smooth non-Archim…

Polarized semiclassical Yau–Tian–Donaldson conjecture. The following are equivalent: